# Average Velocity and Instantaneous Velocity

The average velocity of a particle in a time interval t1 to t2 is defined as its displacement divided by the time interval.

For example – If the particle is at a point A at time t = t1 and at B at time t = t2
The displacement in this time interval is the vector AB.

Then average velocity in this time interval is

$$\vec{v}_{a v}=\frac{\overrightarrow{A B}}{t_{2}-t_{1}}$$

Like displacement, average velocity is also a Vector quantity.

Moreover we can further simplify this equation as from triangle AOB it’s clear that

$$\vec{AB} = \vec{AO} + \vec{OB} \\ \vec{AO} = – \vec{OA} \\ \implies \vec{AB} = vec{OB} – \vec{OA} \\ \vec{OB} = \vec{r_{2}} \\ \vec{OA} = \vec{r_{1}} \\ \implies \vec{AB} = \vec{r_{2}} – \vec{r_{1}} \\ \text{Thus } \vec{v}_{a v}=\frac{\overrightarrow{A B}}{t_{2}-t_{1}} \text{ can be rewritten as } \\ \vec{v}_{a v}= \frac {\vec{r_{2}} – \vec{r_{1}}} {t_{2} – t_{1}}$$

It’s clear from above equation that average velocity takes into consideration only final and initial positions of object and doesn’t care about how object went from initial to final position.

Let’s understand this using an example problem.

Let’s now discuss Instantaneous Velocity

Average Velocity of the particle in a short time interval t to t + Δt is defined as follows.

$$\vec{v_{av}} = \frac {Δ\vec{r}} {Δt}$$

where Δr vector is the displacement in the time interval Δt

Now if we make Δt vanishingly small and find the limiting value of Δr/Δt
This value is Instantaneous Velocity of the particle at time t.

$$\vec{v}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{r}}{\Delta t}=\frac{d \vec{r}}{d t}$$

Let’s understand Instantaneous Velocity using an example.